About the properties of black holes

Published on 09/02/2013

Question:

I have two questions in regards to black holes:

What is the relationship between the size of the star, the size of the black hole, the size of the event horizon, and the size of the general field of influence(area where there is a possibility of being sucked in)? Is there a formula, say, for example, star with mass of 10 tons = black hole with radius of 10 cm = event horizon with radius of 10 kilometers = field of influence with radius of 100 kilometers?

What is the relationship between the size of the black hole and its life span? For example, radius of 10 cm = dissolves in 10 years?

Thank you for your questions about black holes. Some are easier to answer than others. Black holes are described by a mass that is equivalent to the mass of the object from which they formed, so a 10 ton star (which is too light to really exist) would form a 10 ton black hole. Actually, it's not quite that simple, as when a star collapses some of the outer layers could be blown off and escape, but the important point is that the mass of the black hole is determined by the mass of the material that fell into it.

It is difficult to give a size to a black hole, as the theory says that everything is crushed down to an infinitesimally small point. (Some people argue that this is a reason to improving our theory, but the general consensus is still that everything would be crushed down to something so small that it might as well be infinitesimal: about the size of the Planck length, which is about 10-35 m).

We usually define the size of black holes by their event horizon, which is the point of no return: nothing, not even light, can escape from that point. (This is why they are called black holes). The size of the event horizon for a non-spinning black hole is

r = 2GM/c2,

where G is the universal gravitational constant, M is the mass and c is the speed of light. For our 10 ton black hole, that would give r = 1.5 × 10-24 m (depending exactly on your definition of the ton, I've actually used the metric tonne). For a black hole the same mass as the Sun r = 3 km.

Rotating black holes are slightly more complicated, but the numbers are similar, perhaps a factor of two smaller.

Gravity behaves the same for all objects. It gets weaker the further you are away, but there is no clear cut-off point. Therefore, it is difficult to define a field of influence. You will be able to pass closer to black hole and still escape if you are travelling faster. I would suggest that the event horizon is a suitable distance to consider, as this is the point at which there is absolutely no chance of avoiding getting sucked in.

The lifetime of black holes is an interesting questions. We believe that they should lose energy via Hawking radiation. If left on their own, they would eventually radiate away all of their energy and dissipate. The time taken to evaporate is

t = 5120πG2M3/(ħc4),

where π is the mathematical constant, M is the initial mass and ħ is the reduced Planck's constant (often called h-bar). However, in reality astrophysical black holes are not left on their own; they are constantly absorbing background radiation (as well as any gas or other material that gets too close). This means that they gain mass quicker than they radiate it away, so they will live much longer! No black hole that formed from a collapsing star should have evaporated yet, and none will until the Universe is much older. Some may last effectively forever.